Abstract
We propose and solve an optimal vaccination problem within a deterministic compartmental model of SIRS type: the immunized population can become susceptible again, e.g. because of a not complete immunization power of the vaccine. A social planner thus aims at reducing the number of susceptible individuals via a vaccination campaign, while minimizing the social and economic costs related to the infectious disease. As a theoretical contribution, we provide a technical non-smooth verification theorem, guaranteeing that a semiconcave viscosity solution to the Hamilton–Jacobi–Bellman equation identifies with the minimal cost function, provided that the closed-loop equation admits a solution. Conditions under which the closed-loop equation is well-posed are then derived by borrowing results from the theory of Regular Lagrangian Flows. From the applied point of view, we provide a numerical implementation of the model in a case study with quadratic instantaneous costs. Amongst other conclusions, we observe that in the long-run the optimal vaccination policy is able to keep the percentage of infected to zero, at least when the natural reproduction number and the reinfection rate are small.
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Acemoglu, D., Chernozhukov, V., Werning, I., Whinston, M.D.: Optimal targeted lockdowns in a multi-group SIR model. Am. Econ. Rev: Insights. 3(4), 487–502 (2021)
Alvarez, F.E., Argente, D., Lippi, F.: A simple planning problem for COVID-19 lockdown, testing and tracing. Forthcoming on American Economic Review: Insights. 3(3), 367–382 (2021)
Angeli, M., Neofotistos, G., Mattheakis, M., Kaxiras, E.: Modeling the effect of the vaccination campaign on the COVID-19 pandemic. Chaos Solitons Fract 154, 111621 (2022)
Ambrosio, L.: Well Posedness of ODE’s and Continuity Equations with Nonsmooth Vector Fields, and Applications (2017). Rev. mat. Complutense. 30, 427–450 (2017)
Ambrosio, L.: Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158, 227–260 (2004)
Barrett, S., Hoel, M.: Optimal disease eradication. Environ. Dev. Econ. 12(5), 627–652 (2007)
Brito, D.L., Sheshinski, E., Intriligator, M.D.: Externalities and compulsory vaccinations. J. Public Econ. 45, 69–90 (1991)
Calvia, A., Gozzi, F., Lippi, F., Zanco, G.: A simple planning problem for COVID-19 lockdown: a dynamic programming approach. Preprint available online at arXiv:2206.00613 (2022)
Cannarsa, P., Sinestrari, C.: Semiconcave Functions, Hamilton–Jacobi Equations, and Optimal Control. Progress in Nonlinear Differential Equations and their Applications, vol. 58. Birkhäuser, Basel (2014)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)
Federico, S., Ferrari, G.: Taming the spread of an epidemic by lockdown policies. J. Math. Econ. 93, 102453 (2021)
Garriga, C., Manuelli, R., Sanghi, S.: Optimal management of an epidemic: lockdown, vaccine and value of life. J. Econ. Dyn. Control 140, 104351 (2022)
Gatto, N.M., Schellhorn, H.: Optimal control of the SIR model in the presence of transmission and treatment uncertainty. Math. Biosci. 333, 108539 (2021)
Geoffard, P.Y., Philipson, T.: Disease eradication: private versus public vaccination. Am. Econ. Rev. 87(1), 222–230 (1997)
Giaquinta, M., Modica, G.: Mathematical Analysis. An Introduction to Functions of Several Variables. Birkhauser, Basel (2009)
Glover, A., Heathcote, J., Krueger, D.: Optimal Age-Based Vaccination and Economic Mitigation Policies for the Second Phase of the Covid-19 Pandemic. J. Econ. Dyn. Control. 140 (2022)
Goenka, A., Liu, L.: Infectious diseases and endogenous fluctuations. Econ. Theor. 50, 125–149 (2012)
Josephy, M.: Composing functions of bounded variation. Proc. Am. Math. Soc. 83(2), 354–356 (1981)
Hethcote, H.W., Waltman, P.: Optimal vaccination schedules in a deterministic epidemic model. Math. Biosci. 18, 365–381 (1973)
Hritonenko, N., Yatsenko, Y.: Analysis of optimal lockdown in integral economic-epidemic model. Econ. Theor. (2022). https://doi.org/10.1007/s00199-022-01469-7
Ishikawa, M.: Stochastic optimal control of an sir epidemic model with vaccination. In: Proceedings of the 43rd ISCIE International Symposium on Stochastic Systems Theory and its Applications (2012)
Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics, part I. Proc. R. Soc. A 115, 700–721 (1927)
Kruse, T., Strack, P.: Optimal control of an epidemic through social distancing. Preprint available online at https://ssrn.com/abstract=3581295, https://doi.org/10.2139/ssrn.3581295 (2020)
Loertscher, S., Muir, E.V.: Road to recovery: managing an epidemic. J. Math. Econ. 93, 102482 (2021)
Makris, M.: Covid and social distancing with a heterogenous population. Econ. Theor. (2021). https://doi.org/10.1007/s00199-021-01377-2
Miclo, L., Spiroz, D., Weibull, J.: Optimal epidemic suppression under an ICU constraint. J. Math. Econ. 101 (2022)
O’Regan, S.M., Kelly, T.C., Korobeinikov, A., O’Callaghan, M.J.A., Pokrovskii, A.V.: Lyapunov functions for SIR and SIRS epidemic models. Appl. Math. Lett. 23(4), 446–448 (2010)
Rao, I.J., Brandeau, M.L.: Optimal allocation of limited vaccine to control an infectious disease: simple analytical conditions. Math. Biosci. 337, 108621 (2021)
Yong, J., Zhou, X.Y.: Stochastic Control—Hamiltonian Systems and HJB Equations. Springer, Berlin (1999)
Acknowledgements
We thank two anonymous reviewers and the guest editor for valuable comments and suggestions. Salvatore Federico was partially supported by the Italian Ministry of University and Research (MUR), in the framework of PRIN project 2017FKHBA8 001 “The Time-Space Evolution of Economic Activities: Mathematical Models and Empirical Applications”.
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Federico, S., Ferrari, G. & Torrente, ML. Optimal vaccination in a SIRS epidemic model. Econ Theory 77, 49–74 (2024). https://doi.org/10.1007/s00199-022-01475-9
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DOI: https://doi.org/10.1007/s00199-022-01475-9
Keywords
- SIRS model
- Optimal control
- Viscosity solution
- Non-smooth verification theorem
- Epidemic
- Optimal vaccination